cyclically reduced
Let $M(X)$ be a free monoid with involution ${}^{1}$ on $X$. A word $w\in M(X)$ is said to be cyclically reduced if every cyclic conjugate of it is reduced. In other words, $w$ is cyclically reduced iff $w$ is a reduced word and that if $w=uv{u}^{1}$ for some words $u$ and $v$, then $w=v$.
For example, if $X=\{a,b,c\}$, then words such as
$${c}^{1}b{c}^{2}a\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}aba{c}^{2}b{a}^{2}$$ 
are cyclically reduced, where as words
$${a}^{2}bc{a}^{1}\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}c{b}^{2}{b}^{3}c$$ 
are not, the former is reduced, but of the form $a(abc){a}^{1}$, while the later is not even a reduced word.
Remarks. The concept of cyclically reduced words carries over to words in groups. We consider words in a group $G$.

•
If a word is cyclically reduced, so is its inverse^{} and all of its cyclic conjugates.

•
A word $v$ is a cyclic reduction of a word $w$ if $w=uv{u}^{1}$ for some word $u$, and $v$ is cyclically reduced. Clearly, every word and its cyclic reduction are conjugates of each other. Furthermore, any word has a unique cyclic reduction.

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Every group $G$ has a presentation^{} $\u27e8SR\u27e9$ such that

(a)
$R$ is cyclically reduced (meaning every element of $R$ is cyclically reduced),

(b)
closed under^{} inverses (meaning if $u\in R$, then ${u}^{1}\in R$), and

(c)
closed under cyclic conjugation (meaning any cyclic conjugate of an element in $R$ is in $R$).
Furthermore, if $G$ is finitely presented, $R$ above can be chosen to be finite.
Proof.
Every group $G$ has a presentation $\u27e8S{R}^{\prime}\u27e9$. There is an isomorphism^{} from $F(S)/N({R}^{\prime})$ to $G$, where $F(S)$ is the free group^{} freely generated by $S$, and $N({R}^{\prime})$ is the normalizer^{} of the subset ${R}^{\prime}\subseteq F(S)$ in $F(S)$. Let ${R}^{\prime \prime}$ be the set of all cyclic reductions of words in ${R}^{\prime}$. Then $N({R}^{\prime \prime})=N({R}^{\prime})$, since any word not cyclically reduced in ${R}^{\prime}$ is conjugate to its cyclic reduction in ${R}^{\prime \prime}$, and hence in $N({R}^{\prime \prime})$. Next, for each $u\in {R}^{\prime \prime}$, toss in its inverse and all of its cyclic conjugates. The resulting set $R$ is still cyclically reduced. Furthermore, $R$ satisfies the remaining conditions above. Finally, $N(R)=N({R}^{\prime \prime})$, as any cyclic conjugate $v$ of a word $w$ is clearly a conjugate of $w$. Therefore, $G$ has presentation $\u27e8SR\u27e9$.
If $G$ is finitely presented, then $S$ and ${R}^{\prime}$ above can be chosen to be finite sets^{}. Therefore, ${R}^{\prime \prime}$ and $R$ are both finite. $R$ is finite because the number of cyclic conjugates of a word is at most the length of the word, and hence finite. ∎

(a)
Title  cyclically reduced 

Canonical name  CyclicallyReduced 
Date of creation  20130322 17:34:04 
Last modified on  20130322 17:34:04 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 20F10 
Classification  msc 20F05 
Classification  msc 20F06 
Defines  cyclic reduction 
Defines  cyclic conjugation 